A Comprehensive Approach to Exciton Delocalization and Energy Transfer

Electrostatic intermolecular interactions lie at the heart of both the Förster model for resonance energy transfer (RET) and the exciton model for energy delocalization. In the Förster theory of RET, the excitation energy incoherently flows from the energy donor to a weakly coupled energy acceptor. The exciton model describes instead the energy delocalization in aggregates of identical (or nearly so) molecules. Here, we introduce a model that brings together molecular aggregates and RET. We will consider a couple of molecules, each described in terms of two diabatic electronic states, coupled to an effective molecular vibration. Electrostatic intermolecular interactions drive energy fluxes between the molecules, that, depending on model parameters, can be described as RET or energy delocalization. At variance with the standard Förster model for RET and of the exciton model for aggregates, our approach applies both in the weak and in the strong coupling regimes and fully accounts for the quantum nature of molecular vibrations in a nonadiabatic approach. Coupling the system to a thermal bath, we follow RET and energy delocalization in real time and simulate time-resolved emission spectra.


S1 Displaced bosonic creation and annihilation operators
As stated in Sec. 2 (main text), the equilibrium position for each harmonic oscillator varies with the ionicity on the relevant molecule: It is convenient to move the origin of the vibrational coordinate to the equilibrium position, via a Lang-Firsov transformation of the vibrational operators: 1,2 where we introduced displaced bosonic creation and annihilation operators,â i andâ † i .

S2 Rotation of the dimer electronic Hamiltonian on the adiabatic mean field basis
The electronic part of Hamiltonian in Eq. 4 (main text) that describes a pair of dipolar dyes on the diabatic |N 1 , N 2 ⟩, |N 1 , Z 2 ⟩, |Z 1 , N 2 ⟩ and |Z 1 , Z 2 ⟩ basis can be rewritten as follows: where, for each molecule, we define the operators: For each molecule, we define two adiabatic states |G i ⟩ and |E i ⟩, linear combination of the |N i ⟩ and |Z i ⟩: To rotate the Hamiltonian in Eq. 4 on the adiabatic basis for the dimer (|G 1 , G 2 ⟩, |G 1 , E 2 ⟩, |E 1 , G 2 ⟩ and |E 1 , E 2 ⟩), we introduce the following Paulion operators,p . These operators are related to the operators in Eq. 5 as follows: 2,3 By plugging Eqs. 6 and 7 into Eq. 4 and neglecting constant terms, we get: wheren i =p † ip i and M = V /2.
This expression holds true for any choice ofρ 1 andρ 2 value. In order to find the |G i ⟩ and |E i ⟩ basis states that describe the actual ground and excited states of the two molecules in the environment of the dimer (i.e., feeling the potential generated by the other molecule), we set to zero the last two terms in Eq. 9: thus getting: Eq. 9 finally reduces to: which is the Hamiltonian of two non interacting molecules, each one feeling the electric field generated by the other one. Eq. 14 together with Eqs. 10 and 11, give Eq. 8 of the main text, with the mean field transition energies of the dyes reading:

S3 Redfield relaxation tensor
For a given bath and system-bath interaction hamiltonianĤ B andĤ SB (Ĥ bath andĤ dimer−bath in the main text), the terms of the four-dimensional Redfield relaxation tensor R ab,cd read: with the kinetic coefficients reading: where the angle brackets ⟨·⟩ bath indicate the average over the equilibrium bath states and For theĤ bath andĤ dimer−bath defined in the main text, Γ + db,ac and Γ − db,ac read: where q are the db and ac matrix elements of the vibrational operatorŝ Q 1/+(2/−) , two spectral densities are defined as Since the integrals in Eq. 17 span positive frequencies, Eqs. 18 and 19 have to be read as: while for ε a = ε c and ε b = ε d the Redfield kinetic coefficients vanish as we impose I 1/2/+/− (ω = 0) = 0.

S4.1 Debye spectral density
We consider the Debye spectral density: where η measures the strength of the system-bath coupling and ω c is the cut-off frequency.
For homodimers, the η parameter is adjusted as to have at ω + the same value of the   RET velocity depends on the values of the spectral density at frequencies lower than the energy donor vibrational frequency.

S4.2 Centrosymmetric dimers
We always consider systems of aligned molecules, where a positive electrostatic interaction sets the in-phase combination |G 1 E 2 ⟩ + |E 1 G 2 ⟩ at higher energy than the out-of-phase one |G 1 E 2 ⟩ − |E 1 G 2 ⟩, and vice versa for a negative interaction. As discussed in the main text, for two molecules oriented in the same direction (non-centrosymmetric dimers), the bright state reached upon light absorption is the in-phase combination (left part of Fig. 3S reports for reference results also shown in Fig. 2 of the mian text). On the other hand, for molecules oriented in opposite directions, the out-of-phase combination becomes the bright state. The rightmost part of figure 3S shows results for centrosymmetric dimers. More specifically, the systems in the first and third column have the same Hamiltonian (positive electrostatic interactions), but the coherent excitation populates S 2 in the non-centrosymmetric system (first column) and S 1 in the centrosymmetric one (third column), as it can be seen from the populations reported in the middle panels. Moreover, the expectation value ofQ + shows how in the centrosymmetric dimer the coherence is maintained for a longer time respect to the non-censtrosymmetric system, since all the excited state dynamics occurs in the same electronic manifold. The same considerations apply to the systems with negative interactions (second and fourth column). Here the slower relaxation of the centrosymmetric system is due to the higher energy difference between S 1 and S 2 . Figure 3S: Results for dimers of NR (molecular parameters in Tab. 1, main text) with different relative orientations of the monomers (sketches of the dimeric geometries are shown in the top panels; left panels are taken from Fig. 2 in the main text). Top panels: time evolution of the system energy (red). For reference, the energy of the lowest vibronic eigenstates in S 1 and S 2 manifolds are shown as orange and black dotted lines, respectively. Middle panels: time evolution of the populations of the lowest vibronic eigenstate in S 1 and S 2 manifolds (orange and black lines, respectively). Bottom panels: time evolution of ⟨Q + ⟩, and of ∆Q − (green and blue dotted lines, respectively).

S4.3 Steady state optical spectra
We compare the absorption and long time (i.e., after 1 ps) emission spectra obtained from the Liouville-von Neumann dynamics and the steady state spectra calculated through a sum-over-states (SOS) approach. In particular, the Hamiltonian in Eq. 4 in the main text is diagonalized (molecular parameters for Nile Red dimers are reported in the main text) and the absorption spectra are calculated according to: where we assumed that only the lowest vibronic eigenstate is populated at ambient temperature, i runs over the vibronic eigenstates and a lorentzian bandshape with standard deviation σ is associated with every transition. Similarly, once the Kasha's state |f ⟩ is identified, steady state emission spectra are calculated as: where we introduced the Boltzmann distribution to account for the possible thermal population of the excited vibronic eigenstates. Figure 4S shows how the absorption and emission (collected after 1 ps) spectra calculated with the Liouville-von Neumann equation nicely match those obtained using Eqs. 23 and 24.  Figure 5S: Comparison between selective coherent excitation of the energy donor (using onlŷ µ D ; dotted lines) and a monochromatic excitation simulated with a gaussian centered at 2.9 eV with HWHM = 0.1 eV (usingμ D +μ A ; continuous lines) for a DANS-Nile Red pair (molecular parameters reported in the main text) for V =0.02 eV (left panels) and V =0.5 eV (right panels). Top panels: Energy of the system as a function of time; bottom panels: Time evolution of ⟨Q D ⟩ (red lines) and ⟨Q A ⟩ (black lines).

S5 Population fitting for rates extrapolation
Energy transfer rates from the dynamical simulations are obtained fitting the time evolution of the DA * population (eigenstate 106 in all cases) with the equation: where a 3 is the rate of interest.
Fitting parameters relevant to the rates shown in Figs. 6 and 8 of the main text are reported in Tables 1 and 2, respectively.  In Sec. 4.3 in the main text, the total amount of electronic excitation over one of the two molecules of the dimer is evaluated by the following operator: where ρ 1 = ⟨ϕ 1 |Z 1 ⟩⟨Z 1 |ϕ 1 ⟩, |ϕ 1 ⟩ being the ground state of the dimer Hamiltonian (Eq. 4 in the main text).